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Theorem bndss 33715
Description: A subset of a bounded metric space is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
bndss ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑆𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Bnd‘𝑆))

Proof of Theorem bndss
Dummy variables 𝑟 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 metres2 22215 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑆𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆))
21adantlr 751 . . 3 (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆))
3 ssel2 3631 . . . . . . . . . . . . 13 ((𝑆𝑋𝑥𝑆) → 𝑥𝑋)
43ancoms 468 . . . . . . . . . . . 12 ((𝑥𝑆𝑆𝑋) → 𝑥𝑋)
5 oveq1 6697 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑦(ball‘𝑀)𝑟) = (𝑥(ball‘𝑀)𝑟))
65eqeq2d 2661 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
76rexbidv 3081 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (∃𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
87rspcva 3338 . . . . . . . . . . . 12 ((𝑥𝑋 ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))
94, 8sylan 487 . . . . . . . . . . 11 (((𝑥𝑆𝑆𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))
109adantlll 754 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))
11 dfss 3622 . . . . . . . . . . . . . . . . . . 19 (𝑆𝑋𝑆 = (𝑆𝑋))
1211biimpi 206 . . . . . . . . . . . . . . . . . 18 (𝑆𝑋𝑆 = (𝑆𝑋))
13 incom 3838 . . . . . . . . . . . . . . . . . 18 (𝑆𝑋) = (𝑋𝑆)
1412, 13syl6eq 2701 . . . . . . . . . . . . . . . . 17 (𝑆𝑋𝑆 = (𝑋𝑆))
15 ineq1 3840 . . . . . . . . . . . . . . . . 17 (𝑋 = (𝑥(ball‘𝑀)𝑟) → (𝑋𝑆) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
1614, 15sylan9eq 2705 . . . . . . . . . . . . . . . 16 ((𝑆𝑋𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
1716adantll 750 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
1817adantlr 751 . . . . . . . . . . . . . 14 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
19 eqid 2651 . . . . . . . . . . . . . . . . . 18 (𝑀 ↾ (𝑆 × 𝑆)) = (𝑀 ↾ (𝑆 × 𝑆))
2019blssp 33682 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆𝑋) ∧ (𝑥𝑆𝑟 ∈ ℝ+)) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
2120an4s 886 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ (𝑆𝑋𝑟 ∈ ℝ+)) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
2221anassrs 681 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
2322adantr 480 . . . . . . . . . . . . . 14 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
2418, 23eqtr4d 2688 . . . . . . . . . . . . 13 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
2524ex 449 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑋 = (𝑥(ball‘𝑀)𝑟) → 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
2625reximdva 3046 . . . . . . . . . . 11 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) → (∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
2726imp 444 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
2810, 27syldan 486 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
2928an32s 863 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
3029ex 449 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → (𝑆𝑋 → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
3130an32s 863 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑥𝑆) → (𝑆𝑋 → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
3231imp 444 . . . . 5 ((((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑥𝑆) ∧ 𝑆𝑋) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
3332an32s 863 . . . 4 ((((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) ∧ 𝑥𝑆) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
3433ralrimiva 2995 . . 3 (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) → ∀𝑥𝑆𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
352, 34jca 553 . 2 (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) → ((𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆) ∧ ∀𝑥𝑆𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
36 isbnd 33709 . . 3 (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)))
3736anbi1i 731 . 2 ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑆𝑋) ↔ ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋))
38 isbnd 33709 . 2 ((𝑀 ↾ (𝑆 × 𝑆)) ∈ (Bnd‘𝑆) ↔ ((𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆) ∧ ∀𝑥𝑆𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
3935, 37, 383imtr4i 281 1 ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑆𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Bnd‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  cin 3606  wss 3607   × cxp 5141  cres 5145  cfv 5926  (class class class)co 6690  +crp 11870  Metcme 19780  ballcbl 19781  Bndcbnd 33696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-mulcl 10036  ax-i2m1 10042
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-rp 11871  df-xadd 11985  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-bnd 33708
This theorem is referenced by:  ssbnd  33717
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